# CosIntegral

CosIntegral[z]

gives the cosine integral function .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• .
• CosIntegral[z] has a branch cut discontinuity in the complex z plane running from - to 0.
• For certain special arguments, CosIntegral automatically evaluates to exact values.
• CosIntegral can be evaluated to arbitrary numerical precision.
• CosIntegral automatically threads over lists.
• CosIntegral can be used with Interval and CenteredInterval objects. »

# Examples

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## Basic Examples(6)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Asymptotic expansion at Infinity:

Asymptotic expansion at a singular point:

## Scope(36)

### Numerical Evaluation(5)

Evaluate numerically to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate CosIntegral efficiently at high precision:

CosIntegral threads elementwise over lists and matrices:

CosIntegral can be used with Interval and CenteredInterval objects:

### Specific Values(3)

Value at a fixed point:

Values at infinity:

Find a local maximum as a root of :

### Visualization(2)

Plot the CosIntegral function:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(8)

CosIntegral is defined for all positive real values:

Complex domain:

CosIntegral is not an analytic function:

Nor is it meromorphic:

CosIntegral is neither non-decreasing nor non-increasing:

CosIntegral is not injective:

CosIntegral is not surjective:

CosIntegral is neither non-negative nor non-positive:

It has both singularity and discontinuity in (-,0]:

CosIntegral is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the derivative:

### Integration(3)

Indefinite integral of CosIntegral:

Definite integral of CosIntegral over its entire real domain:

More integrals:

### Series Expansions(3)

Taylor expansion for CosIntegral around :

Plots of the first three approximations for CosIntegral around :

Find series expansion at infinity:

CosIntegral can be applied to power series:

### Function Identities and Simplifications(4)

Use FullSimplify to simplify expressions containing the cosine integral:

Use FunctionExpand to express CosIntegral through other functions:

Simplify expressions to CosIntegral:

Argument simplifications:

### Function Representations(5)

Primary definition of CosIntegral:

Series representation of CosIntegral:

CosIntegral can be represented in terms of MeijerG:

CosIntegral can be represented as a DifferentialRoot:

## Generalizations & Extensions(1)

Find series expansions at infinity:

## Applications(6)

Average radiated power for a thin linear half-wave antenna:

Plot the imaginary part in the complex plane:

Plot the logarithm of the absolute value in the complex plane:

Solve a differential equation:

Real part of the EulerHeisenberg effective action:

Find the leading term in :

Plot Nielsen's spiral:

The curvature is a simple function of the parameter:

## Properties & Relations(7)

Use FullSimplify to simplify expressions containing the cosine integral:

Use FunctionExpand to express CosIntegral through other functions:

Find a numerical root:

Obtain CosIntegral from integrals and sums:

Obtain CosIntegral from a differential equation:

Calculate the Wronskian:

Laplace transform:

## Possible Issues(2)

CosIntegral can take large values for moderatesize arguments:

A larger setting for \$MaxExtraPrecision can be needed:

## Neat Examples(1)

Nested integrals:

Wolfram Research (1991), CosIntegral, Wolfram Language function, https://reference.wolfram.com/language/ref/CosIntegral.html (updated 2022).

#### Text

Wolfram Research (1991), CosIntegral, Wolfram Language function, https://reference.wolfram.com/language/ref/CosIntegral.html (updated 2022).

#### CMS

Wolfram Language. 1991. "CosIntegral." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/CosIntegral.html.

#### APA

Wolfram Language. (1991). CosIntegral. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CosIntegral.html

#### BibTeX

@misc{reference.wolfram_2024_cosintegral, author="Wolfram Research", title="{CosIntegral}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/CosIntegral.html}", note=[Accessed: 22-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_cosintegral, organization={Wolfram Research}, title={CosIntegral}, year={2022}, url={https://reference.wolfram.com/language/ref/CosIntegral.html}, note=[Accessed: 22-May-2024 ]}